Publications

Refereed papers

Cartan Connections and Atiyah Lie Algebroids

J. Attard, J. François, S. Lazzarini, and T. Masson

Journal of Geometry and Physics148103541(2020)

abstract

This work extends previous developments carried out by some of the authors on Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan connections in a framework relying on two Atiyah Lie algebroids based on a $H$-principal fiber bundle $\mathcal{P}$ and its associated $G$-principal fiber bundle $\mathcal{Q} := \mathcal{P} \times_H G$, where $H \subset G$ defines the model for a Cartan geometry. The first main result of this study is a commutative and exact diagram relating these two Atiyah Lie algebroids, which allows to completely characterize Cartan connections on $\mathcal{P}$. Furthermore, in the context of gravity and mixed anomalies, our construction answers a long standing mathematical question about the correct geometrico-algebraic setting in which to combine inner gauge transformations and infinitesimal diffeomorphisms.

Given a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold and $\nabla$ a covariant derivative on $V$, let $P = -(\lvert g \rvert^{-1/2} \nabla_\mu \lvert g \rvert^{1/2} g^{\mu\nu} u \nabla_\nu + p^\mu \nabla_\mu +q)$ be a nonminimal Laplace type operator acting on smooth sections of $V$ where $u,\,p^\nu,\,q$ are $M_N(\mathbb{C})$-valued functions with $u$ positive and invertible.
For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} \,a \,e^{-tP} \sim_{t \downarrow 0} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written as an integral of the functions $a_r(a, P)(x) = \text{tr}\,[a(x) \,\mathcal{R}_r(x)]$.
This paper revisits the previous computation of $\mathcal{R}_2$ by the authors and is mainly devoted to a computation of $\mathcal{R}_4$. The results are presented with $u$-dependent operators which are universal (i.e. $P$-independent) and which act on tensor products of $u$, $p^\mu$, $q$ and their derivatives via (also universal) spectral functions which are fully described.

Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P = - [g^{\mu\nu} u(x)\partial_\mu\partial_\nu + v^\nu(x)\partial_\nu + w(x)]$ where $u,\,v^\nu,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr}(a e^{-tP}) \underset{t \downarrow 0^+}{\sim} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written locally as $a_r(a, P)(x) = \text{tr}[a(x) \mathcal{R}_r(x)]$. The computation of $\mathcal{R}_2$ is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.

For an elliptic selfadjoint operator $P =-[u^{\mu\nu}\partial_\mu \partial_\nu +v^\nu \partial_\nu +w]$ acting on a fiber bundle over a Riemannian manifold, where $u^{\mu\nu},v^\mu,w$ are $N\times N$-matrices, we develop a method to compute the heat-trace coefficients $a_r$ which allows to get them by a pure computational machinery. It is exemplified in any even dimension by the value of $a_1$ written both in terms of $u^{\mu\nu}=g^{\mu\nu}u,v^\mu,w$ or diffeomorphic and gauge invariants. We also address the question: when is it possible to get explicit formulae for $a_r$?

We sum up known results on the inclusion of diffeomorphisms in a gauge theory so as to obtain the BRST algebra of a Einstein-Yang-Mills theory. We then show the compatibility of this operation with the (so-called) dressing field method which allows a systematic reduction of gauge symmetries. The robustness of the so obtained scheme is illustrated on the geometry of General Relativity and on the richer example of the second-order conformal structure.

Residual Weyl symmetry out of conformal geometry and its BRS structure

J. François, S. Lazzarini, and T. Masson

Journal of High Energy Physics20159(2015)

abstract

The conformal structure of second order in m-dimensions together with the so-called (normal) conformal Cartan connection, is considered as a framework for gauge theories. The dressing field scheme presented in a previous work amounts to a decoupling of both the inversion and the Lorentz symmetries such that the residual gauge symmetry is the Weyl symmetry. On the one hand, it provides straightforwardly the Riemannian parametrization of the normal conformal Cartan connection and its curvature. On the other hand, it also provides the finite transformation laws under the Weyl rescaling of the various geometric objects involved. Subsequently, the dressing field method is shown to fit the BRS differential algebra treatment of infinitesimal gauge symmetry. The dressed ghost field encoding the residual Weyl symmetry is presented. The related so-called algebraic connection supplies relevant combinations found in the literature in the algebraic study of the Weyl anomaly.

In the last few years, the so-called Chen et al. approach of the nucleon spin decomposition has been widely discussed and elaborated on. In this letter we propose a genuine differential geometric understanding of this approach. We mainly highligth its relation to the ``dressing field method'' we advocated in [C. Fournel, J. François, S. Lazzarini, T. Masson, Int. J. Geom. Methods Mod. Phys. 11, 1450016 (2014)]. We are led to the conclusion that the claimed gauge-invariance of the Chen et al. decomposition is actually unreal.

Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{A},\mathcal{H},\mathcal{D})$ on the crossed product $G \ltimes_{\alpha, \text{red}} \mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^*$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold.

International Journal of Geometric Methods in Modern Physics1111450016(2014)

abstract

In this paper we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this systematic approach.

A regular spectral triple is proposed for a two-dimensional $\kappa$-deformation. It is based on the naturally associated affine group $G$, a smooth subalgebra of $C^*(G)$, and an operator $\mathcal{D}$ defined by two derivations on this subalgebra. While $\mathcal{D}$ has metric dimension two, the spectral dimension of the triple is one. This bypasses an obstruction described in [Iochum, Masson, Schücker, and Sitarz, Compact $\kappa$-deformation and spectral triples, Reports on Mathematical Physics, 68(1):37--64, 2011], on existence of finitely-summable spectral triples for a compactified $\kappa$-deformation.

In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to ordinary and to similar noncommutative gauge theories.

The aim of the paper is to answer the following question: does $\kappa$-deformation fit into the framework of noncommutative geometry in the sense of spectral triples? Using a compactification of time, we get a discrete version of $\kappa$-Minkowski deformation via $C^*$-algebras of groups. The dynamical system of the underlying groups (including some Baumslag--Solitar groups) is used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation.

We construct discrete versions of $\kappa$-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical system of the underlying discrete groups (which include some Baumslag--Solitar groups) is heavily used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation. The dimension of these spectral triples is unrelated to the number of coordinates defining the $\kappa$-deformed Minkowski spaces.

Connections on Lie algebroids and on derivation-based non-commutative geometry

S. Lazzarini and T. Masson

Journal of Geometry and Physics62387--402(2012)

abstract

In this paper we study some generalized notions of connections on transitive Lie algebroids from an algebraic point of view. Differential calculi are introduced to manage connections $1$-forms and curvature $2$-forms. Two examples are study in details: the Atiyah Lie algebroid of a principal fiber bundle and the space of derivations of the endomorphisms algebra of a $SL(n)$-vector bundle. Using these two examples we show that the notion of generalized connections studied here is strongly related to the notion of connections on the derivation-based non-commutative geometry of this algebra of endomorphisms. As such, relative to ordinary connections, generalized connections on an Atiyah Lie algebroid is the same kind of generalization as (derivation-based) non-commutative connections.

We introduce the new notion of $\varepsilon$-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of $\varepsilon$-graded Lie algebras. We define and study the associated notion of $\varepsilon$-derivation-based differential calculus, which generalizes the derivation-based calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with the examples of $\varepsilon$-graded matrix algebras. Finally, we apply this formalism to a graded version of the Moyal algebra, for which we recover the recently constructed candidate for a renormalizable gauge action on Moyal space supplemented by terms built from a scalar field and a 2-covariant symmetric tensor field.

Derivations of a (noncommutative) algebra can be used to construct various consistent differential calculi, the so-called derivation-based differential calculi. We apply this framework to the noncommutative Moyal algebras for which all the derivations are inner and analyse in detail the case where the derivation algebras generating the differential calculus are related to area preserving diffeomorphisms. The ordinary derivations corresponding to spatial dimensions are supplemented by additional derivations necessarely related to additional covariant coordinates. It is shown that these latter have a natural interpretation as Higgs fields when involved in gauge invariant actions built from the noncommutative curvature. The UV/IR mixing problem for (some of) the resulting Yang-Mills-Higgs models is discussed. A comparition to other noncommutative geometries already considered in the litterature is given.

This is an extended version of a communication made at the international conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a $SU(n)$-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes.

The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall state is examined. Standard Chern-Simons anyons as well as ``non standard'' anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective ability to stabilize attractive Bose gases under fast rotation in the thermodynamical limit is studied. Stability can be obtained for standard anyons while for non standard anyons, stability requires that the range of the corresponding statistical interaction does not exceed the typical wavelenght of the atoms.

The Born-Infeld lagrangian for non-abelian gauge theory is adapted to the case of the generalized gauge fields arising in non-commutative matrix geometry. Basic properties of static and time dependent solutions of the scalar sector of this model are investigated.

In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions usually introduced to classify ordinary invariant connections, and we expand them using algebraic objects coming from the noncommutative setting. The main result is that the classification can be performed using a ``reduced'' algebra, an associated differential calculus and a module over this algebra.

We consider Maxwell-Chern-Simons models involving different non-minimal coupling terms to a non relativistic massive scalar and further coupled to an external uniform background charge. We study how these models can be constrained to support static radially symmetric vortex configurations saturating the lower bound for the energy. Models involving Zeeman-type coupling support such vortices provided the potential has a ``symmetry breaking'' form and a relation between parameters holds. In models where minimal coupling is supplemented by magnetic and electric field dependant coupling terms, non trivial vortex configurations minimizing the energy occur only when a non linear potential is introduced. The corresponding vortices are studied numerically.

Quantum Hall conductivity in a Landau type model with a realistic geometry II

F. Chandelier, Y. Georgelin, T. Masson, and J.-C. Wallet

Annals of Physics314476(2004)

abstract

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractionnal quantization of the Hall conductivity depending on the value of $NB/2\pi$ ($N$ is the number of charge carriers and $B$ is the magnetic field). When $NB/2\pi$ is irrationnal, we show that monovalued wave functions can be constructed only on the graph of a free group with two generators. When $NB/2\pi$ is rationnal, the relevant space becomes a puncturated Riemann surface. We finally discuss our results from a phenomenological viewpoint.

We present a new non-abelian generalization of the Born-Infeld Lagrangian. It is based on the observation that the basic quantity defining it is the generalized volume element, computed as the determinant of a linear combination of metric and Maxwell tensors. We propose to extend the notion of determinant to the tensor product of space-time and a matrix representation of the gauge group. We compute such a Lagrangian explicitly in the case of the $SU(2)$ gauge group and then explore the properties of static, spherically symmetric solutions in this model. We have found a one-parameter family of finite energy solutions. In the last section, the main properties of these solutions are displayed and discussed.

Quantum Hall conductivity in a Landau type model with a realistic geometry

F. Chandelier, Y. Georgelin, T. Masson, and J.-C. Wallet

Annals of Physics30560-78(2003)

abstract

In this paper, we revisit some quantum mechanical aspects related to the Quantum Hall Effect. We consider a Landau type model, paying a special attention to the experimental and geometrical features of Quantum Hall experiments. The resulting formalism is then used to compute explicitely the Hall conductivity from a Kubo formula.

We propose a construction of a global phase diagram for the quantum Hall effect. This global phase diagram is based on our previous constructions of visibility diagrams in the context of the Quantum Hall Effect. The topology of the phase diagram we obtain is in good agreement with experimental observations (when the spin effect can be neglected). This phase diagram does not show floating.

We consider a class of $(2+1)$-dimensional nonlocal effective models with a Maxwell-Chern--Simons part for which the Maxwell term involves a suitable nonlocality that permits one to take into account some $(3+1$)-dimensional features of ``real'' planar systems. We show that this class of models exhibits a hidden duality symmetry stemming from the Maxwell-Chern-Simons part of the action. We discuss and illustrate this result in the framework of a $(2+1)$-dimensional effective model describing (massive) vortices and charges with realistic interactions.

e analyze various properties of the visibility diagrams that can be used in the context of modular symmetries and confront them to some recent experimental developments in the Quantum Hall Effect. We show that a suitable physical interpretation of the visibility diagrams which permits one to describe successfully the observed architecture of the Quantum Hall states gives rise naturally to a stripe structure reproducing some of the experimental features that have been observed in the study of the quantum fluctuations of the Hall conductance. Furthermore, we exhibit new properties of the visibility diagrams stemming from the structure of subgroups of the full modular group.

We construct a family of holomorphic $\beta$-functions whose RG flow preserves the $\Gamma(2)$ modular symmetry and reproduces the observed stability of the Hall plateaus. The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependance of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occuring in the crossover of the permitted transitions is discussed.

On the noncommutative geometry of the endomorphism algebra of a vector bundle

T. Masson

Journal of Geometry and Physics31142(1999)

abstract

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.

We study the noncommutative differential geometry of the algebra of endomorphisms of any $SU(n)$-vector bundle. We show that ordinary connections on such $SU(n)$-vector bundle can be interpreted in a natural way as a noncommutative $1$-form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.

We consider the action of the modular group $\Gamma (2)$ on the set of positive rational fractions. From this, we derive a model for a classification of fractional (as well as integer) Hall states which can be visualized on two ``visibility'' diagrams, the first one being associated with even denominator fractions whereas the second one is linked to odd denominator fractions. We use this model to predict, among some interesting physical quantities, the relative ratios of the width of the different transversal resistivity plateaus. A numerical simulation of the tranversal resistivity plot based on this last prediction fits well with the present experimental data.

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.

We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivation-based differential calculus introduced by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these definitions.

A recently proposed definition of a linear connection in non-commutative geometry, based on a generalized permutation, is used to construct linear connections on $GL_q(n)$. Restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed. Candidates for generalized permutation on $GL_q(n)$ are found. It is shown that, for a given generalized permutation, there exists one and only one associated linear connection. Properties of the linear connection are discussed, in particular its bicovariance, torsion and commutative limit.

We apply a recently proposed definition of a linear connection in non commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there exists a non trivial family of linear connections only when the two parameters obeys a specific relation.

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.

A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.

A note about the comparison of the WCA and the Self-consistent WCA perturbation methods

L. Kazandjian and T. Masson

Journal of Chemical Physics992275(1993)

abstract

The self‐consistent WCA method is generally regarded as more accurate than the WCA method at high densities, which may seem surprising in view of the orders of approximation expected. It is shown in this note that, in fact, the relative ranking of the two methods depends on the representation chosen for the indirect correlation function of the reference hard‐sphere fluid.

Lecture given at the 6th Peyresq meeting 'Integrable systems and quantum field theory'

abstract

This informal introduction is an extended version of a three hours lecture given at the 6th Peyresq meeting ``Integrable systems and quantum field theory''. In this lecture, we make an overview of some of the mathematical results which motivated the development of what is called noncommutative geometry. The first of these results is the theorem by Gelfand and Neumark about commutative $C^\ast$-algebras; then come some aspects of the $K$-theories, first for topological spaces, then for $C^\ast$-algebras and finally the purely algebraic version. Cyclic homology is introduced, keeping in mind its relation to differential structures. The last result is the construction of the Chern character, which shows how these developments are related to each other.

The dressing field method of gauge symmetry reduction, a review with examples

J. Attard, J. François, S. Lazzarini, and T. Masson

Foundations of Mathematics and Physics one Century After Hilbert, New Perspectives, J. Kouneiher (Ed.), Springer, 2018

abstract

Gauge symmetries are a cornerstone of modern physics but they come with technical difficulties when it comes to quantization, to accurately describe particles phenomenology or to extract observables in general. These shortcomings must be met by essentially finding a way to effectively reduce gauge symmetries. We propose a review of a way to do so which we call the dressing field method. We show how the BRST algebra satisfied by gauge fields, encoding their gauge transformations, is modified. We outline noticeable applications of the method, such as the electroweak sector of the Standard Model and the local twistors of Penrose.

This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common mathematical pattern on which the paper concludes.

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features.

Discontinued papers

Local description of generalized forms on transitive Lie algebroids and applications

C. Fournel, S. Lazzarini, and T. Masson

arxiv 1109.4282v1

(2011)

abstract

In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex with a Leray-Serre spectral sequence. We apply the general theory to Atiyah Lie algebroids and to derivations on a vector bundle.

A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model

T. Masson and J.-C. Wallet

arxiv 1001.1176

(2010)

abstract

In this paper we consider the Spontaneous Symmetry Breaking Mechanism (SSBM) in the Standard Model of particles in the unitary gauge. We show that the computation usually presented of this mechanism can be conveniently performed in a slightly different manner. As an outcome, the computation we present can change the interpretation of the SSBM in the Standard Model, in that it decouples the $SU(2)$-gauge symmetry in the final Lagrangian instead of breaking it.

We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies of composite bosons to describe 2D harmonically trapped fast rotating Bose gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for $\nu$ (ratio of particle to vortex numbers) are $\nu = \frac{p}{q}$ ($p$, $q$ are any integers) with even product $pq$, including numerically favored values previously found and predicting a richer set of values. We show that those values can be understood from a bosonic analog of the law of the corresponding states relevant to the electronic FQHE. A tentative global phase diagram for the bosonic system for $\nu <1$ is also proposed.